Re: Rate problem

Just in case a picture helps...

... where (key in spoiler) ...

Spoiler:

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case t), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

Re: Rate problem

By seeing that you are trying to differentiate a composite function of t with respect to t. And the chain rule says this will involve differentiating with respect to the whole of the inner function of t. (D is the inner function of t.)

Re: Rate problem

We choose to represent the function V(t) as V(D(t)). The chain rule says that .

Properly speaking, where something came from is not a mathematical question. The solution to this question is a sequence of statements, each of which is either true or false (hopefully, true). The only proper question is why a certain statement is true.

Edit: Yes, it's legitimate to ask metamathematical questions, such as what reasoning one used to come up with a certain solution. Hopefully, post #8 answered this question.

Re: Rate problem

It makes more sense to use the Chain rule now that I see there is a function V(D(t)). I did not know this was the function that I was supposed to use to solve the problem. The issue I am having is how to formulate the problem so I can solve it. I did not know to express the volume as the diameter when the time changes the diameter...I am still a bit confused as to how I can get this, but I think I will skip this one for now and come back to it later because I have other problems I need to finish as well as other classes.

Re: Rate problem

Originally Posted by daigo

It makes more sense to use the Chain rule now that I see there is a function V(D(t)). I did not know this was the function that I was supposed to use to solve the problem.

There is never a function that your are supposed to use to solve a problem- look at all functions and formulas that have anything to do with the problem.
In this case you are told that "a snowball melts in volume @ 1 cc/min" so the rate of change of V with respect to t is surely relevant.

The issue I am having is how to formulate the problem so I can solve it. I did not know to express the volume as the diameter when the time changes the diameter...I am still a bit confused as to how I can get this, but I think I will skip this one for now and come back to it later because I have other problems I need to finish as well as other classes.